# Full text of "Treatise On Analysis Vol-Ii"

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384       XV    NORMED ALGEBRAS AND SPECTRAL THEORY

PROBLEMS

1. Let A be the algebra with involution ^gW °f continuous bounded complex-valued
functions on R, and let /x be a bounded positive measure on R, with support equal to R.
Show that

is a bitrace on A for which the prehilbert space A/ng = A is separable, but that the star
algebra tfg is not separable (Section 7.4, Problem 4).

2.   Let A be Lebesgue measure on R and let A be the subalgebra with involution

consisting of square-A-integrable functions. Then g(x, y) = J x(t)y(t) dX(t) is a bitrace

on A satisfying (U) and (N), but the measure mg is not bounded, and the right-hand
side of (15.9.3) is not defined for all x e A. If B c A is the subalgebra of ^£(R) consisting
of A-integrable functions, then

is a positive linear form on B satisfying (15.9,3) but the measure mg is not bounded.

3.   Let A be the subalgebra with involution of ^C(I)» where I = [0, 1 ], consisting of func-
tions x with continuous second-order derivative on I and such that x(0) = 0. Show that

/(*)= Cx(t)dt +

Jo

is a positive linear form on A, such that the corresponding bitrace g satisfies the con-
dition (U) but not the condition (N). (If (/,,) is a sequence of functions belonging to
^C(I), with values in [0, 1] and continuous second derivatives, and such that/n(0 = 1
in a neighborhood of 0 and/n(0 — 0 for t^.l/n, consider in the Hilbert space Hff the
sequence of functions xn e A such that xn(t) — //„(/).)

4.    Let A be the subalgebra with involution of #C(I) consisting of continuously differenti-
able functions on I which vanish at 0. Show that the formf(x) — x'(0) on I is a positive
linear form for which the corresponding bitrace g is zero (and therefore satisfies (U)
and (N)) but does not satisfy (B).

5.    Let A be the algebra with involution (the involution being x\-. >x) of complex- valued
functions on [0, 1] of the form P(f, log f), where P is a polynomial in two variables with

r1

complex coefficients. Then the linear form f(x) =     x(t) dt is positive, and satisfies

Jo
(B) because A has a unit element; the corresponding bitrace g satisfies (N) but not (U).

6.    Let F be a set endowed with an associative law of composition (x,y)\-+xy and a
neutral element e, and let x\-+x* be an involution on F (i.e., a bijection of F onto F
such that e* « e, (x*)* = x and (xy)* = y*x*). A representation of F in a Hilbertllows in all cases.